Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps I
Abstract
In this series of articles, we analyse the level-sets of length functions on the moduli space of compact hyperbolic surfaces of fixed genus. This work ultimately culminates in a proof that typical hyperbolic surfaces have an optimal spectral gap. In this first article, we introduce new volume functions , counting the expected number of closed geodesics of type and length on a random hyperbolic surface of genus . So far, this function has only been considered in the case where the type is simple, in which case it can be expressed as a combination of Weil-Petersson volumes polynomials, as proven by Mirzakhani. We provide an integral expression for for any prescribed type , which we use to prove that admits a full asymptotic expansion in powers of . We then claim that the coefficients in this expansion, as a function of the length variable , belong to a newly-introduced class of functions called "Friedman-Ramanujan functions". We relate this claim to the study of the spectral gap of the Laplace-Beltrami operator, and prove it when fills a surface of Euler characteristic or , providing a method to explicitly compute all coefficients in the second-order expansion. We conclude by displaying how the presence of tangles (which is an event of vanishing probability) prevents the sum over all types to satisfy the Friedman-Ramanujan property at the second order.
Cite
@article{arxiv.2304.02678,
title = {Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps I},
author = {Nalini Anantharaman and Laura Monk},
journal= {arXiv preprint arXiv:2304.02678},
year = {2026}
}
Comments
71 pages, 18 figures. This new version is shorter due to making the former last section into a standalone article (to appear on arxiv). Content is otherwise unchanged